These students are already looking for a closed rule and have done substantial investigation, with diagrams as their aid. They are probing possibilities, but have a clear goal of finding the general rule.

Question: How does the teacher respond to the student idea to "Work backward"?

Although it may not have been clear to the teacher whether this would be an effective strategy, she lets the students proceed down this path, using guiding questions to make sure they understand what strategies they are using. As it turns out, working backwards, per se, does not offer insights, but does prompt them to rearrange the diagram in a way that does inspire new thinking.

Question: How does the teacher guide them through the later stages of problem solving?

After the students have found a helpful representation and are closer to deriving the rule, she asks them to check that the rule is reasonable and they discover a missing step, dividing by two. She also asks them to look at two cases and to go back to the table they worked with earlier, encouraging them to explore different representations of the problem.